Differential Equations Study Sheet Page 3
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101.3
Mixing Problems
dQ
= Rate In - Rate Out.
dt
Consider a tank that initally contains 50 gallons of pure water. A salt solution contain-
ing 2 pounds of salt per gallon of water is poured into the tank at a rate of 3 gal/min.
The solution leaves the tank also at 3 gal/min.
Therefore Input = 2(lb/gal)*3(gal/min).
Output = ?(lbs/gal)*3(gal/min).
Q(t)
Salt in Tank =
.
50
Q(t)
Therefore output of salt =
(lbs/gal)*3(gal/min).
50
dQ
Q(t)
= Rate In - Rate Out = 2 lbs/gal*3gal/min -
(lbs/gal)*3(gal/min).
dt
50
3Q
6 lbs/min -
lbs/min.
50
Solve via seperation of Variables.
1.4
Existance and Uniqueness
dy
Given
= f (t, y). If f is continuous on some interval, then there exists at least one
dt
solution on that interval.
∂
If both f (t, y) and
f (t, y) are continuous on some interval then an initial value
∂y
problem on that interval is guaranteed to have exactly one Unique solution.
1.5
Phase Lines
Takes all the information from a slope fields and captures it in a single vertical line.
Draw a vertical line, label the equilibrium points, determine if the slope of y is positive
or negative between each equilibrium and label up or down arrows.
1.6
Classifying Equilibria and the Linearization Theorem
Source: solutions tend away from an equilibrium
f (y
) > 0.
o
Sink: solutions tend toward an equilibrium
f (y
) < 0.
o
Node: Nither a source or a sink
f (y
) = 0 or DNE.
o
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